Let \({C}, {D}, {E}\)  be categories.
Recall the notion of an adjunction of two variables \({C}\times {D}\rightarrow {E}\)  (see, for example, [1]}).
We will denote an adjunction of two variables only by its tensor functor \(\otimes \colon {C}\times {D}\rightarrow {E}\) .
If \({E}\)  has pushouts, then there is an induced pushout product, or box product on morphisms:
given morphisms \(f \colon A \rightarrow B\)  in \({C}\)  and \(g \colon X \rightarrow Y\)  in \({D}\) , their pushout product (relative to \(\otimes \) ) is the induced morphism in \({E}\)  given by
\(\begin{tikzcd}A \otimes Y \underset{A \otimes B}{\sqcup } B \otimes X[r, "f \square g"]& B \otimes Y\,.\end{tikzcd}\) 
